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Chapter 7: Problem 30

Evaluate the geometric series. $$ \sum_{m=5}^{91}(-2)^{m} $$

Short Answer

Expert verified
The sum of the given geometric series is \(-8,200,534\). This is calculated using the formula for the sum of a geometric series, with \(a_1 = -32\), \(r = -2\), and \(n = 87\) terms in the series.

Step by step solution

01

Identify the first term and the common ratio

The given geometric series is: \[ \sum_{m=5}^{91}(-2)^{m} \] The first term of the series is when \(m=5\), so: \[ a_1 = (-2)^5 = -32 \] The common ratio is what the term is multiplied by to get the next term. In this series, each term is multiplied by \(-2\). So, the common ratio is: \[ r = -2 \]
02

Calculate the number of terms in the series

The index of the series runs from \(5\) to \(91\), inclusive. To find the total number of terms in the series, we can use: \[ n = 91 - 5 + 1 = 87 \]
03

Apply the geometric series sum formula

The formula for the sum of a geometric series is given by: \[ S_n = \frac{a_1(r^n - 1)}{r - 1} \] Plugging the values for our series: \[ S_n = \frac{-32((-2)^{87}-1)}{-2 - 1} \]
04

Evaluate the sum

Now we can compute the sum: \[ S_n = \frac{-32((-2)^{87}-1)}{-3} \] Using a calculator, we get: \[ S_n = -8200534 \] So, the sum of the given geometric series is \(-8,200,534\).

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(a) Evaluate \(\left(\begin{array}{l}9 \\ 3\end{array}\right)\). (b) Evaluate \(\left(\begin{array}{l}9 \\ 6\end{array}\right)\).

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